"why, sometimes I've believed as many as 6 impossible things before breakfast" said the White Queen

The first thought that struck me when I glanced at the morning paper - What is betting all about really? Is there a mathematical formula that can garuntee a win??

Probability theory says p = n/N, where p is the probability of an event to occur, n is the number of times that it can occur, and N is the number of trials, or total number of events that can occur.

Now lets consider the event of drawing a particular number "x" from a given set of N numbers. Probability that you will draw "x" is p=1/N , given ofcourse that the N numbers are all different with no repetitions.

The larger N gets, the lower is the probability of drawing "x". Conversely, the probability of NOT drawing "x" (lets call it p') gets higher. For example, if N=100, p' = 99/100 % or 99%. And if N=100000, p'=99.999%

So we can conclude here that the most sensible thing would be to call p as probability that the drawn number will be within a subset of N. It is winning within a number of consecutive drawings.

So here, we can say that the median of these drawings M = 1-(1-p)^N, and here, the median would represent the degree of certainity of winning. If N = 692, degree of certainity (DC) = 49.96% (close to 50%)

Following this , we can conclude that
N =[ log(1-DC)]/log(1-p)
So, if u know that the probability of a win is p, and you are willing to take chances on a degree of certainity of x%, then by using the formula you will be able to calculate the number of consecutive trials.
If Dc=1, it will lead to a mathematical error. From here, we can see that to expect a degree of certainity of 1 (or 100% certainity that the event will surely occur) is absurd.

Nothing comes with absolute certainty, but to a degree of certainty! That's mathematics, and that's the only TRUTH

P.S - The title of this article is a line from Lewis Carroll's famous classic - "Through the Looking glass".

- Posted by Anindita Chatterjee (Quantinuum member)